Journal of Public Economics 130 (2015) 45–58

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Journal of Public Economics

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Equilibrium tax rates and income redistribution: A laboratory study☆

Marina Agranov, Thomas R. Palfrey ⁎Division of the Humanities and Social Sciences, California Institute of Technology, Mail Code 228-77, Pasadena, CA 91125, United States

☆ The financial support of the National Science FoundaSage Foundation, and the Gordon and Betty Moore Foacknowledged. The paper benefited from discussionconference and seminar presentations. We are especiasuggestions from Ernesto Dal Bo, John Londregan, Thomaand editor. Kirill Pogorelskiy provided excellent research⁎ Corresponding author.

E-mail addresses: magranov@hss.caltech.edu (M. Agra(T.R. Palfrey).

1 See, for example, McCarty et al. (2006).

http://dx.doi.org/10.1016/j.jpubeco.2015.08.0080047-2727/© 2015 Elsevier B.V. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 24 November 2014Received in revised form 7 August 2015Accepted 16 August 2015Available online 22 August 2015

Keywords:VotingTaxesMeltzer–RichardExperiment

This paper reports results from a laboratory experiment that investigates the Meltzer–Richard model of equilib-rium tax rates, inequality, and income redistribution. The experiment varies the amount of wage inequality andthe political process used to determine tax rates. We find that higher inequality leads to higher tax rates; the ef-fect is significant and large inmagnitude. The tax rates and labor supply functions are both quantitatively close tothe theory. The result is robust to the political institution. The theoretical model of Meltzer–Richard is extendedto incorporate social preferences in the formof altruism and inequity aversion,which are found to havenegligibleeffects in the data.

© 2015 Elsevier B.V. All rights reserved.

1. Introduction

In the US and other democratic countries, taxes are decided by ademocratic political process, and income tax policy in particular hasenormous redistributive consequences. Much of the expenditures thatare financed by income taxes are either almost entirely redistributive,such as Food Stamps or Aid to Families with Dependent Children, orhave significant redistributive components, such as subsidies toeducation (college loans, head start, work study), public transit, andhealth insurance. These expenditures are generally aimed at benefitinglower incomemembers of society, while the costs of these programs areborne in proportion to income (or, under progressive taxation, morethan proportionally to income). However, standard economic analysisimplies that, unless the elasticity of labor supply with respect to after-tax wages is zero for all individuals, this redistribution comes at a cost.Thus, on the one hand, income taxes reduce inequality, which is gener-ally regarded to be a positive improvement to society, but on the otherhand, taxes may negatively affect efficiency of the economy throughdistortions in the labor market. This fundamental equity-efficiencytradeoff drives much of the political debate and polarization over eco-nomic policy, which is considered by most political scientists to be theprimary dimension of political competition in modern democracies.1

tion (SES-0962802), the Russellundation (1158) is gratefullyand commentary at several

lly grateful for comments ands Romer, as well as the refereesassistance.

nov), trp@hss.caltech.edu

There is now a rather well developed and rigorous, equilibrium-based theory addressing the positive question of how the level of in-come taxes are determined in the democratic society, starting withthe work of Romer (1975), Roberts (1977) and Meltzer and Richard(1981). These models are based on the median voter theory developedby Black (1958) and Downs (1957).2 The equity–efficiency tradeoff inthese models is captured by a distortion to labor supply created by agap between the after-tax wage and a worker's marginal productivity.The heterogeneity in the agents' productivities is the driving force be-hind inequality in the pre-tax incomes in these models, as it is in themodel we study in the present paper.While the theoretical implicationsof these models have potentially enormous economic consequences,both in terms of inequality level in society and economic efficiency, asan empirical matter, these theories are extremely difficult to test usingmacro field and historical data sets.

Not only is such data relatively limited, but there are open method-ological issues about the extent to which these studies enable one todraw causal conclusions, as well as the deeper problem of endogeneityof the economic and political variables using historical or contemporarydata. For example, one basic implication of these median voter modelsof tax policy is that, all else equal, greater pre-tax inequality will leadto higher taxes. At the same time the model predicts that, all elseequal, higher taxes will lead to a decline in aggregate output. Besidescausality issues, it is hard to pin down exactly which policies areredistributive, or more precisely, howmuch redistribution is associatedwith various policies. Moreover, the key variables, inequality, taxes, and

2 The present paper focuses primarily on behavioral and positive questions about thepolitical–economic equilibrium that determines tax policy, rather than normative con-cerns about optimal tax rates. Thus we explore a different set of questions than is ad-dressed in the literature on incentive efficient tax schemes, pioneered by Mirlees (1971).

3 This mechanism has been used in tax referendums in the U.S. See Holcombe (1977)and Holcombe and Kenny (2007,2008).

4 There is also abundant evidence from laboratory experiments suggesting a role of di-rect preferences over redistribution, or social preferences, in economic decisionmaking inenvironmentswhere inequality plays a role. See, for example, Andreoni andMiller (2002),Bolton and Ockenfels (2000), Fehr and Schmidt (1999), Fisman et al. (2007), Palfrey andRosenthal (1988).

46 M. Agranov, T.R. Palfrey / Journal of Public Economics 130 (2015) 45–58

income are all endogenous and causally intertwined. For cross-nationalstudies, political institutions vary across countries, and in none of thesystems are tax rates determined by “pure” majority rule vote. Ratherthere are a variety of ways of deciding taxes, ranging from decisionsmade by elected representatives to highly decentralized systems thatmore closely resemble referenda.

There have been a number of careful studies that acknowledge thesedifficulties and attempt to overcome them. Unfortunately, taken collec-tively, these studies have led to ambiguous, and sometimes conflictingconclusions. Several studies attempt to test the median voter tax hy-pothesis, which states that the tax rate and/or government expendi-tures in democracies will correspond to the ideal level of publicexpenditure of the median voter. Meltzer and Richard (1983) test thiswith data on their categorization of redistributive expenditures in theU.S. between 1936 and 1977, excluding expenditures on public goodssuch as public safety, defense, and infrastructure. They don't find directevidence for the hypothesis, but find that purely redistributive expendi-tures are positively correlatedwith the ratio ofmean tomedian income.Milanovic (2000), in a cross-sectional study of 24 democracies, alsofinds that income redistribution to the poor correlates with measuresof income inequality, but finds little support for the median voter hy-pothesis. On the other hand, Perotti (1996), in his cross-sectionalstudy of 67 countries, does not find significant evidence for a positiverelationship between inequality and middle class tax rates. Thus, theoverall picture is one of mixed empirical findings. While some of thefindings are suggestive of a link that would be consistent with the me-dian voter hypothesis, the link is tenuous and does not help identifythemechanism bywhich themedian voter preferences are implement-ed in the political process.

Experiments offer a valuable tool for advancing our understandingof the political economy of redistribution and taxation by providing aclean test of the theoretical models in very simple environments,while preserving key incentives and tradeoffs that people face outsideof the laboratory. Data created from a carefully controlled setting canbe used toward the development of better models. Our experimentcan be seen as one of the first attempts to study the interaction betweenlabor market and political behavior in the laboratory, while keeping allthe remainingdetails (political institution and distribution of productiv-ities) constant and varying one parameter at a time.While the environ-ment in the laboratory eliminates many of the confounding factors thatare present in themore complex phenomena of fiscal policymaking andlabor supply in large economies, experiments have a significantadvantage over empirical research using historical time-series orcross-sectional data in the evaluation of the theoretical models thateconomists develop in order to better understand these complex phe-nomenon. At the same time, although the experiment we conduct pro-vides a sharp test of the basic theory, the existence of those manyconfounding factors makes us reluctant to make any grand claimsabout voting over redistributive taxes inmass electorates and legislativebodies or aggregate labor supply responses to income tax rates in largeeconomies.

The experimentwe report in this paper explores questions about theequity–efficiency tradeoff vis-a-vis redistributive taxation, the equilibri-um effect of wage inequality on income tax rates, and themedian voterhypothesis about the political economy consequences of voting overtaxes. Our laboratory environment is designed to correspond to theMeltzer–Richardmodel. The individuals participating in our experimentoperate in two interconnected environments: a political environment,where the level of taxation is determined, and a labormarket (economicenvironment), in which, given an income tax schedule, individuals withvarying wage rates choose labor supply that generates pre-tax income.Because of the redistributive effect of income taxation and because indi-viduals differ in their productivities and hence their incomes, individ-uals in our experiment have different indirect preferences for the levelof taxation and these preferences depend upon the distribution of pro-ductivities in the economy. Political institutions are themeans bywhich

these heterogeneous preferences are aggregated into a public decisionon the tax rate. However, because the tax rate in turn affects the amountof income that is generated by the private economy, agents' preferencesfor redistribution themselves are endogenous and depend on aggregatelabor supply responses to taxes.

The experiment is motivated by three distinct considerations. Theprimary motivation was summarized above: the large empirical litera-ture devoted to studying these questions about the equity–efficiencytradeoff and in particular the median voter hypothesis that impliesgreater inequality leads to higher taxes, has not succeeded in comingto any consensus about any of the important questions raised by thetheoretical political models of redistributive taxation. Our experimentcan address these theoretical issues by providing data from a simple en-vironment where preferences, technology, and the political process aretightly controlled, leading to sharp theoretical predictions. By exoge-nously controlling the level of inequality, we can address the causalquestion of how the degree of inequality in the economy affects thelevel of redistributive taxation.

A second consideration concerns the role of the specific set of insti-tutions that implements democratic outcomes. One of the shortcomingsof the classic political economy models of income taxation is that theyare completely silent about the mechanics of the political process bywhich a tax rate is chosen. The models simply assume that the taxrate preferred by the median voter will emerge, as if by an invisible po-litical hand. To address this, our experimental design compares the taxrates that emerge under two canonical majoritarian political processesthat correspond to much different extensive form games: direct democ-racy and representative democracy. In the direct democracy mechanism,themedian voter's preferred policy is elicited directly,3 while in the rep-resentative democracy system voters choose in an election betweentwo office-motivated candidates who compete by choosing tax ratesas their platforms.

A third consideration concerns the potentially important effects ofdirect preferences for redistribution. The standard political economy ap-proach described above characterizes indirect preferences for redistri-bution, based on the assumption that voters are completely selfish.There is a substantial empirical literature on direct preferences for redis-tribution, largely addressing questions of cross-cultural differences inpreferences for equality, tolerance of inequality, or interdependent pref-erences (see Alesina and Giuliano (2011)). 4 Redistributive taxation isan environment where social preferences can naturally influence be-havior and outcomes. With this in mind, we extend the basic model tocharacterize the equilibrium effects of social preferences, and test forsuch effects in the data.

We have three main results, which map directly back to the threemotivations described above. The first result is that the implementedtax rates in the experiment closely track the preferences of the medianproductivity worker, providing support for the median voter theory ofequilibrium tax rates. Higher tax rates lead to lower aggregate laborsupply and lower total income, and the observed voting decisions indi-cate that the preferences of voters over tax rates take into account thisequity–efficiency tradeoff. As a consequence, we find that higher in-equality leads to significantly more income redistribution throughhigher taxation.

The secondmain finding is that the first result is robust to the specif-ic political institution used to set the tax rates: we observe very similarbehavior and outcomes under direct democracy and representative

5 Theoretical predictions for two alternative models of social preferences are derived inSection 5.2.

6 Equivalently, taxes are used to finance a level of public good according to the linear

technology,y ¼ 1n∑

n

j¼1t �wjx j, and all agents value the public good according to the function

V(y) = y, which corresponds to the last term of Eq. (1).

47M. Agranov, T.R. Palfrey / Journal of Public Economics 130 (2015) 45–58

democracy. Observation of choice behavior of voters in each politicalmechanism (direct and representative democracy) allows us to backout estimates of the revealed-preferred ideal points of different votertypes, and in both institutions, the estimated ideal tax rates are mono-tone in productivity as predicted by the theory, with less productive(low wage) individuals preferring higher taxes.

The thirdmain finding is that direct preferences for redistribution, inthe formof either altruism or inequity aversion, affect neither individuallabor supply decisions nor the implemented tax rates in our data. Thequantitative measures of observed labor supply and the average taxrates in the experiment are not significantly different from the standardequilibriummodel with selfish preferences. We construct a model withhomogeneous preferences to estimate the altruism and inequity aver-sion parameters, using the data on labor supply and voting decisions.The estimation fails to reject the hypothesis that the social preferenceparameters are equal to zero, implying that social preferences apparent-ly do not play an important role in either labor supply responses totaxes, or preferences over redistribution, at least in our laboratoryenvironment.

In the remainder of this section,we discuss some related experimen-tal literature. Section 2 presents the equilibriummodel of redistributivetaxation (with selfish preferences), which serves as the theoreticalfoundation for the main hypotheses about labor supply functions andimplemented tax rates for the experiment. Section 3 presents the exper-imental design and procedures. The results are presented in Sections 4and 5.

1.1. Related literature

There is an extensive experimental literature in economics aimed atmeasuring preferences for redistribution. Some studies abstract awayfrom efficiency considerations and focus on self-interest versus fairness(e.g. Forsythe et al. (1994)), while more recent papers incorporate effi-ciency by varying exogenously the size of the total pie (Andreoni andMiller (2002), Fisman et al. (2007)). Bolton and Ockenfels (2006) reportseries of voting games, inwhich subjects are confrontedwith two distri-butions of incomes: one that promotes efficiency and a second thatpromotes equity. Tyran and Sausgruber (2006) offer evidence that in-equality averse social preferences may explain voting behavior overnon-distortionary redistribution, in an experiment where subjectswere endowed with one of two different income levels and vote on afixed amount of redistribution. Hochtl et al. (2012) show that the abilityof inequity aversion to explain voting behavior on redistribution maydependon thepre-tax distribution of income. In all thepapers describedabove, the amount of resources to be distributed is fixed exogenouslyand participants can only decide how to reallocate this surplus. In ourexperiments subjects' labor market decisions determine the totalsurplus generated, so both the total size of the pie and the distributionof income is endogenous.

Four recent studies are more closely related to our paper. Konradand Morath (2011) is the only other experiment that is motivated bythe Meltzer–Richard model. Both the focus and the methodology aredifferent. They study how prospects of incomemobility may affect pref-erences for redistributive taxes in an individual decision-making exper-imentwithout strategic interaction between subjects. In particular, eachhuman subject is paired with two computers who choose actions thatmaximize their own earnings; human subjects are aware of the com-puters' strategies. In the one treatment without mobility observed taxrates are in line with theoretically predicted ones. The important meth-odological difference in their study is the use of computerized agents inplace of strategic interaction between human subjects. Esarey et al.(2012) study preferences for redistribution in a real effort experimentbut address a much different question. They focus on the linkagebetween these preferences and ideological positions elicited from asurvey. Preferences over tax rates are elicited by using the medianvoting rule to select group tax rate. The authors find that preferences

for redistribution were largely driven by self-interest. Durante et al.(2014) investigate how preferences for redistribution vary with socialpreferences, risk aversion, self-interest and the source of pre-tax in-equality. The main finding is that subjects' preference for redistributiondecreases substantially when the initial distribution of endowments isdetermined based on the task performance (earned income) ratherthan randomly (luck). Grosser and Reuben (2013) report two experi-ments where subjects earn income in a double action market. In thefirst experiment trading profits are redistributed according to one ofseveral exogenously fixed rules. The goal is to see whether equal-share redistribution affects trading efficiency. In a competitive equilibri-um there should be no such effect in theory, and they observed onlysmall effects. In the second experiment, redistribution is endogenousand determined by candidate competition as in our study. Because thetaxes are non-distortionary and the median voter has low income, thetheoretical equilibrium tax rate is 100%, which is close to what isobserved.

2. Model and theoretical predictions

In this sectionwe lay out the primitives of themodel and derive equi-librium under the assumption that agents are purely selfish.5 The econ-omy consists of n N 1 agents. Agents operate in a perfectly competitiveand frictionless labor market and also participate in a democratic politi-cal process that determines taxes which in turn affect labor decisions.

We start by discussing the decision problem of an agent in the labormarket assuming that the tax rate is fixed. Then we characterize themajority rule equilibrium tax rate, by deriving the induced preferencesof voters, assuming rational expectations about how aggregate laborsupply responds to changes in the income tax rate.

2.1. The labor market

Agent i is endowed with productivity wi. Individuals are identical inall other respects. The difference in choice of labor and consumptionarises solely because of the differences in productivity. An agent withproductivity wi who supplies xi units of labor earns pre-tax incomeyi = wixi and bears an effort cost of 1

2 x2i which represents the tradeoff

between labor and leisure. Income and costs are measured in units ofconsumption. In addition, each agent pays a fraction t of earned incomein taxes. Tax revenues are redistributed in equal shares.6 Thus the payoffUi of agent i consists of three parts: after-tax disposable income, cost oflabor, and an equal share of collected taxes, where the latter depends onthe entire profile of productivities, w = (w1, …, wn) and labor supplydecisions x = (x1, …, xn):

Ui wi; xi; tð Þ ¼ 1−tð Þ �wixi−12x2i þ

1n

Xnj¼1

t �wjxj: ð1Þ

Given the tax rate t, agent i chooses labor supply xi that maximizes(1) above, taking x-i as given. The utility function is concave, and theunique optimal labor supply for individual i is characterized by thefirst order condition:

x�i wi; tð Þ ¼ 1−n−1n

t� �

wi: ð2Þ

Thus, all productive agents (i.e.,wi N 0) have positive labor supply forall tax rates, t ∈ [0, 1]. Labor supply is declining in the tax rate and is

Table 1Parameters and equilibrium tax rates.

48 M. Agranov, T.R. Palfrey / Journal of Public Economics 130 (2015) 45–58

proportional to a worker's productivity. Hence, pre-tax income isproportional to the square of productivity.

2.2. Equilibrium tax rates under majority rule

We next derive the indirect preferences of agents over tax rates, andcharacterize the majority rule equilibrium.

The equilibrium payoff of agent iwhen the tax rate t is implementedand all other agents follow the behavior prescribed by the equilibriumin the labor market is:

U�i wi; tð Þ ¼ 1

21−tð Þ2− t2

n2

� �w2

i þtn

1−n−1n

t� �

Z ð3Þ

where Z ¼ ∑n

j¼1w2

j denotes the aggregate income of the economy if the

tax rate is t = 0.Our first result, Proposition 1, characterizes preferences of agents

over tax rates and derives the majority rule equilibrium tax rate.

Proposition 1. Agents' preferences over tax rates satisfy the followingproperties7:

1. Single-peakedness: for any wi, there exists ti⁎ ∈ [0, 1] such thatUi⁎(wi, t) b Ui

⁎(wi, tV) for all t b t' ≤ ti⁎ and Ui⁎(wi, t) b Ui

⁎(wi, t')for all ti⁎ ≥ tVN t

2. Ideal points are ordered by productivity: ti⁎ ≤ tj⁎ ⇔ wi N wj

3. The majority rule equilibrium tax rate, tm⁎ , is given by the ideal taxrate of the median productivity worker:

t�m ¼n2

n2−1�

1nZ−w2

m

2nþ 1

Z−w2m

if w2m ≤

1nZ

0 if w2m N

1nZ

2666664

: ð4Þ

The intuition for this characterization is straightforward. Agentswith lower productivity prefer higher taxes, because they enjoy sub-stantial redistributive benefits which for the most part come from thetax payments of the higher productivity, and hence higher income,agents. In contrast, agents with higher productivity prefer lower taxes(or no taxes at all), because they end up subsidizing the large portionof the tax revenues from which they receive back only a small part inbenefits. Specifically, voters with below average income prefer positivetax rates, while voters with above average income prefer zero tax rates.

Single-peakedness and monotonicity of ideal tax rates with respectto productivities, combined with the majority rule, imply that theagent with the median productivity (median voter) is decisive. Put dif-ferently, the tax rate specified in Eq. (4), which is the tax rate most pre-ferred by the median voter, is the unique tax rate that is majoritypreferred to any other tax rate, and is therefore a Condorcet winner.This result echoes the median voter theorem from the spatial model ofelectoral competition.

Notice that total income in equilibrium is ∑ ni¼1 U�

i ðwi; t Þ ¼12 ð1− ðn−1Þ2

n2 t2Þ∑ni¼1w

2i and it is maximized when t = 0 since taxes

are distortionary.

7 The proof is in Appendix A. These properties are central in the theoretical literaturethat studies the political economy of redistributive taxation. Romer (1975) assumes thatagents have Cobb–Douglas preferences over consumption and leisure and derives condi-tions under which the preferences of agents are single-peaked in the tax rate. Roberts(1977) derives amore general condition that guarantees that ideal points are inversely or-dered by income. Meltzer and Richard (1981) assume the regularity condition of Roberts(1977).

A natural next question that arises in this setup is: How do tax ratescompare across economies that differ in the distribution of productivitylevels of its agents? The following corollary to Proposition 1 provides ananswer to this question.

Corollary. Consider two economies with n individuals, which differonly in the profile of productivities: wA in economy A and wB ineconomy B, and suppose that wm

A = wmB . Then, t⁎A = t⁎B = 0 if

and only if w2mN

1n Z

AN 1n Z

B , t⁎A = t⁎B = 0 if and only if 1n Z

ANw2mN

1n Z

B

and t⁎A N t⁎B N 0 if and only if 1n Z

AN 1n Z

BNw2m.

The corollary can be interpreted in terms of inequality in productiv-ities as measured approximately by the variance of worker productiv-ities. To see this, notice that in the special case where the medianproductivity equals the mean productivity, 1n Z is approximately equalto the variance of wi, with the approximation being arbitrarily closefor large n. In this case, an increase in the variance that leaves themean unchanged will lead to a higher equilibrium tax rate. The taxrate chosen by the median voter will be higher in the economy inwhich the productivity levels are more unequal as captured by thisvariance-related measure, 1n Z. Also, if the distribution of productivities

in economy A is more skewed than the one in economy B then 1n Z

AN 1n Z

B

and we would expect (weakly) higher taxes in economy A than ineconomy B. The intuition for this result comes from the fact thattax revenues are rebated back to all agents in equal shares. Whenhigher productivity agents become more productive, they supplymore labor and, thus, contribute more to the total tax revenues.Therefore, the median voter would prefer higher taxes and more re-distribution since an increase in the tax rebate associated with an in-crease in tax rates outweighs the decrease in after-tax disposableincome.

3. Experimental design

Our design has two different treatment dimensions. The first dimen-sion varies the level of wage (productivity) inequality among the agentsin the economy in order to test one of the main predictions of thetheoretical model, that greater inequality leads to more redistributivetaxation.We have two distributional treatments, which we call Low in-equality andHigh inequality. The productivity of themedian voter is thesame in both treatments (wm

Low = wmHigh), but the relevant inequality

measure is higher in High than Low (ZLow b ZHigh). Both have interiorequilibrium tax rates, with 0 b t⁎ Low b t⁎ High b 1.

Table 1 specifies the values used in each treatment and lists the idealtax rates for all agents, assuming selfish preferences. In both of the dis-tributional treatments, there are five individuals, each with a differentwage rate. The only difference between parameters in the high andlow inequality treatments is the productivity of the most productiveagent.

The second dimension of the design varies the political mechanismfor implementing a tax rate.We consider two very different competitivedemocratic institutions for determining the tax rate. As explained in theintroduction, the motivation for looking at two different institutions isthat the theory does not specify an extensive form for the majoritarian

High inequality treatment Low inequality treatment

Agent Productivity Ideal tax rate Agent Productivity Ideal tax rate

2 0.62 1 2 0.626 0.59 2 6 0.54

10 0.53 3 10 0.2814 0.37 4 14 0.0035 0.00 5 18 0.00

The median voter's ideal point is indicated in bold font in the table.

Table 2Experimental design.

Regime High inequality Low inequality

DD 2 sessions (60 subjects; 12 groups) 3 sessions (70 subjects; 14 groups)RD 2 sessions (49 subjects; 7 groups) 2 sessions (49 subjects; 7 groups)

49M. Agranov, T.R. Palfrey / Journal of Public Economics 130 (2015) 45–58

process. Because there are many possible “democratic” mechanisms inpractice, it is important to see if the results depend on the institutionaldetails, or if the results are robust across different mechanisms. In aworld with perfect information and perfect optimization by all agents,the subgame perfect equilibrium in both regimes theoretically couldproduce the same tax rate outcome, whichwill correspond to themedi-an voter ideal point. The two institutions we use in the experiment aredirect democracy and representative democracy. The two institutionswere designed such that themedian voter's ideal tax rate is the outcomeof the unique subgame perfect equilibrium in both regimes. Details aregiven in Section 3.1 below.

3.1. Experimental procedures

All the experiments were conducted at the CASSEL (California SocialScience Experimental Laboratory) using students from the University ofCalifornia, Los Angeles. Subjects were recruited from a database of vol-unteer subjects.8 Nine sessions were run, using a total of 228 subjects.No subject participated in more than one session. We used a betweensubjects design, so each subject participated in only one treatment.Table 2 summarizes the sessions.

The experimental currency was called tokens. Each token a subjectearned was converted to dollars at an exchange rate of $ 1 = 200tokens.9 Total earnings for a subject was the sum of earnings across allperiods in the session, plus a $10 show up fee. Average earnings, includ-ing the show up fee, were approximately $32with a standard deviationof $7.8. Sessions lasted approximately 2 h on average.

Upon arrival to the laboratory, subjects were divided into groupsof five or seven agents: five in the DD sessions and seven in the RDsessions. Five subjects in each group performed the role of agentsand two additional subjects in the RD sessions performed the roleof the candidates. Each agent in a group was assigned one of thefive productivities (see Table 1). Productivity assignments and thegroup assignments were fixed for the whole duration of the session.At the very beginning of the session each agent was told their ownproductivity, but also told the productivity of each of the other fouragents.

There were two parts in each session. In the first part, which lastedfor 10 periods, subjects gained experience with the labor market. Inthe second part of the experiment, which also lasted for 10 periods, de-pending on the session subjects participated in either the DD or the RDgame. Instructions for the second part of the session were given to theparticipants only after they finished the first part.10 We will nowdescribe the specific experimental procedures that were common toall the sessions and then describe how different political regimes wereimplemented.

In the first part of a session, at the beginning of each period agentswere informed of the tax rate for that period. Then they chose howmuch labor to supply without knowing what other subjects in theirgroup chose.11 Labor supply decisions were allowed to be any numberbetween 0 and 25 with up to two decimal places.12 After all five agents

8 The software for the experiment was developed from the open source Multistagepackage, available for download at http://software.ssel.caltech.edu/.

9 The exchange rate was higher ($ 1=100 tokens) for the low inequality treatment be-cause the potential theoretical earnings were lower.10 Appendix B contains the instructions for the DD High inequality treatment.11 The terminology in the experiment avoided reference to work, effort, productivity orother terms associatedwith labormarkets. The individual labor supply decisionwas calledthe “investment level” and productivities/wages were called “values”. pre-tax labor in-come was called “investment earnings”.12 Recall that the optimal choice of labor given the tax rate isxiðwi; tÞ ¼ ð1− n−1

n tÞ �wi ¼ð1−0:8tÞ �wi . Thus, for all agents and for all tax rates, the theoretically optimal choice oflabor is away from the boundaries (strictly below 25 and strictly above 0), except forthe agent with highest productivity in High inequality treatment (wi = 35). Agent withwi=35 should choose xi(35, t) = 25 for any tax rate below 0.375. In equilibrium, the up-per bound of 25 is not binding for either parameter set.

had made their choice, subjects received feedback that specified thelabor supply of each agent in their group, and an agent's own payoffwas displayed on the screen, broken down into three parts: after-tax in-come, the quadratic cost of labor, and their tax rebate (equal share ofcollected taxes). After the period was over, the group moved on to thenext period which was identical to the previous one except for thetax rate imposed at the beginning of the period. In this trainingpart of the session, subjects went through different possible taxrates, in the following order: 0.50, 0.15, 0.70, 0.62, 0.35, 0.05, 0.27,0.75, 0.90, and 0.20.

To help subjects calculate hypothetical earnings from different laborsupply choices, they were provided with a built-in calculator that ap-peared on their monitors. To use the calculator, subjects had to entertwo numbers: a labor supply decision and a guess for the total taxescollected from the other members in their group. Then, the calculatorcomputed the payoff of the subject in this hypothetical scenario takinginto account the current tax rate in this training period and the wageassigned to the subject.

3.1.1. Experimental protocol specific for direct democracyIn the second part of the DD sessions, at the beginning of each period

each agent was asked to submit a proposal for the tax rate. The medianproposal (third lowest tax rate) was announced to all subjects and im-plemented in that period. 13After the tax rate was determined, subjectschose their labor supply as in the first training part. Again, after the taxrate was determined, subjects could use the on-screen calculator toevaluate different hypothetical scenarios before they submitted theirlabor supply decision. This two-stage process was repeated 10 times(10 periods).

3.1.2. Experimental protocol specific for representative democracyRepresentative democracy (RD) is implemented as Downsian candi-

date competition, by introducing two additional players into the game,both of whom are purely office-motivated candidates, with no privatepreferences over tax rates. This leads to a three stage game. In the firststage, the two candidates simultaneously submitted tax rate proposals.In the second stage, voters observed the two candidates' tax rate pro-posals and voted for one of the candidates, with no abstention. The taxrate proposal submitted by the candidate who received a majority ofvotes was implemented for that period. In the third stage, the processwas the same as in the DD sessions: agents observed the tax rate,chose how much to work and then got feedback for that period. Theonly source of earnings for the candidates in the last 10 periods waswinning elections: thewinning candidate in a period earned 200 tokensand the loser earned 0 tokens. This payoff structure aimed to incentivizecandidates to propose the ideal tax rate of the median voter, since, intheory, it defeats any other proposed tax rate if all agents are choosingtheir labor supply decisions optimally. As in the DD regime, once thetax rate for the periodwas determined, agents could use the built-in cal-culator to evaluate hypothetical scenarios before submitting the finallabor decision.14

13 It iswell known (Moulin, 1980) that under thismechanismevery voter has a dominantstrategy to propose his or her ideal tax rate.14 Additional RD sessions were also conducted with an alternative protocol that wasproblematic because it eliminated the learning phase and limited comparability with theDD sessions. Those sessions exhibited slower convergence to the theoretically predictedtax rates, but were otherwise similar.

Table 3Implemented tax rates.

High inequality, t⁎ = 0.53 Low inequality, t⁎ = 0.28

Mean (st err) Median Mean (st err) Median

Implemented taxes 0.50 (0.03) 0.55 0.26 (0.03) 0.25

Note. Robust standard errors are in parentheses, clustered by group.

Fig. 1. Implemented taxes, dynamics.

50 M. Agranov, T.R. Palfrey / Journal of Public Economics 130 (2015) 45–58

The first (training) 10 periods of the RD sessions were the same asin the DD sessions except that the two candidates were also given atask. In order to focus the candidates' attention during these periods,in each period each candidate was randomly assigned one of theagents, was told the agent's productivity and the tax rate for that pe-riod, and then was asked to guess the labor supply of that agent. Acandidate earned 100 tokens for guessing correctly and 0 tokensfor guessing incorrectly, where the correct guess was defined aswithin 2 points of the actual labor supply decision of that agent inthat period. At the end of each period, the candidates observed allthe labor choices of all five agents in their group.

4. Results

In this section we test the main predictions of the theoreticalmodel presented in Section 2 combining the data from both polit-ical regimes. We investigate the median voter hypothesis, theeffect of inequality on redistribution, aggregate labor market be-havior, and the equity-efficiency tradeoff. In Section 5 we extendthe analysis of results by comparing behavior and outcomesunder both DD and RD regimes and testing for the effects of socialpreferences.

4.1. Implemented taxes

The theoretical results of Section 2 imply the hypothesis that greaterinequality leads to higher taxes. Hence, we should observe higher taxrates in our high inequality treatment than in our low inequality treat-ment. The data strongly support this hypothesis. Table 3 presents sum-mary statistics of implemented taxes in each inequality treatment.15

Fig. 1 shows the evolution of the average implemented tax for thesame part of the game.

Table 3 shows that taxes are higher in the high than in the lowinequality treatment, and the effect is highly significant. This re-sult is also confirmed statistically by regressing the implementedtax rates on a dummy variable for the High inequality treatment.The estimated coefficient (0.24) is positive and highly significant(p = 0.00).16

The second prediction of the theory is the median voter hypothe-sis, that for both inequality treatments the ideal tax rate of the medi-an productivity agent will be implemented. Our data provide supportfor this hypothesis. On average, in both inequality treatments taxesconverge to the ones predicted by the theory almost exactly (seeFig. 1). This is confirmed statistically for each inequality treatment

15 Throughout Section 4, we pool the data from the DD and RD treatments and focus onthe last 10 periods of each session, which is the portion of the data where the tax rates aredetermined endogenously by either the DD or RD mechanism. The more extended dataanalysis in Section 5 will show that pooling across the two institutional treatments isjustified.16 In fact, as Fig. 1 clearly shows, in every single period tax rates are higher in the highinequality treatment than in the low inequality treatment. Using a Wilcoxon rank-sumtest performedperiod-by-period, the distribution of taxes in the high inequality treatmentis significantly different from the low inequality treatment in 9 out of 10 periods at the 5%significance level (the exact p-values are reported in Table 11 in Online Appendix 3).

separately, based on the means and standard errors reported inTable 3.

Result 1. In both inequality treatments the implemented tax rates arenot significantly different from the ideal tax rate of the medianproductivity worker, as given in Proposition 1. Thus we find that taxrates are significantly higher when inequality is high.

4.2. Labor supply

Table 4 reports the mean difference between actual labor choices ofagents and the predicted ones, broken down by productivity level andinequality treatment. The data show that behavior of agents in thelabor market is close to that predicted by theory. For each tax rate andeach subject, the predicted value is obtained from Eq. (2). For the highinequality treatment we have 19 groups and therefore have 190observations for each productivity level; for the low inequality treat-ment we have 21 groups and 210 observations for each productivitylevel. As reflected in the table, there is a general pattern of oversup-ply by low wage workers and undersupply by high wage workers.Most of these differences are small in magnitude and insignificant,with one clear exception. The highest wage workers (w = 35) inthe high inequality treatments on average supplied labor by nearly3 units below the theoretical optimum. In contrast, for the highestproductivity agents (w = 18) in the low inequality treatments thisundersupply was of a much smaller magnitude and not significantlydifferent from zero.

To estimate the labor supply functions of the agents, we define the

normalized labor supply function, L(t), as: LðtÞ ≡ x�i ðwi ;tÞwi

¼ 1− n−1n t .

Table 5 reports the Tobit estimates obtained by regressing observednormalized labor supply ð xiwi

Þ on a constant and the tax rate. We do

Table 4Mean differences between observed and predicted labor supply.

High inequality Low inequality

Productivity 2 0.59 (0.38) 0.30 (0.24)Productivity 6 0.09 (0.13) 0.00 (0.18)Productivity 10 0.34 (0.34) 0.32 (0.18)Productivity 14 0.18 (0.12) 0.04 (0.24)Productivity 18 −0.40 (0.37)Productivity 35 −2.67* (1.11)

Note. Robust standard errors are in parentheses, clustered by subject. *p b 0.05.

Table 5Estimated normalized labor supply functions.

Productivity a p-Value b p-Value

2 1.04 (0.22) 0.86 −0.37 (0.45) 0.336 0.98 (0.05) 0.69 −0.73 (0.10) 0.5010 0.99 (0.03) 0.73 −0.69 (0.09) 0.2014 0.99 (0.02) 0.79 −0.76 (0.05) 0.4118 0.96 (0.05) 0.45 −0.73 (0.13) 0.6135 (t N 0.36) 0.97 (0.12) 0.79 −0.83 (0.19) 0.8635 (t b 0.36) 0.64 (0.08) 0.41 −0.42* (0.20) 0.04

Note. Robust standard errors in the parentheses, clustered by subject. *p b 0.05.

51M. Agranov, T.R. Palfrey / Journal of Public Economics 130 (2015) 45–58

this separately for each productivity level, pooling across the two in-equality treatments. Because we have 40 groups and 10 observationsper group, this gives us 400 observations for each of the four lower pro-ductivity levels is (which are the same in both high and low inequalitytreatments) and 190 observations for w = 35 agents and 210 observa-tions for the w = 18 agents. For the highest productivity worker inthe high inequality treatment (wi=35), the constraint xi ≤ 25 is bindingif the tax rate is sufficiently low (t ≤ 0.375). So we run separate regres-sions for t ≤ 0.375 and t N 0.375 for this one class of worker-voters.Thus, the table reports the estimates of the constant term a and thecoefficient on the tax rate b for a total of seven different regressions. Ac-cording to the theoretical normalized labor supply equation derived forselfish agents, the estimates for the first six (unconstrained) regressionsare predicted to be a=1 andb ¼ − n−1

n ¼ −0:8. For the constrained re-gression reported in the last row, the predicted estimateswere a=0.71and b = 0.

The results reported in Table 5 are largely consistent with the find-ings from Table 4 and very close to the prediction labor supplyin Eq. (2). The estimated constant terms and coefficients are notsignificantly different from the predicted ones at the 5% level with oneexception. The one exception is the estimated slope of the response tothe tax rate for the highest productivity agent (w = 35) when con-straint xi ≤ 25 is binding. The estimated slope is significantly negative,which reflects the undersupply of labor by the highest productivityworkers, as also reported in the last row of Table 4.

Result 2. Labor supply decisions by agents are approximately optimaland consistent with the theoretical labor supply functions given inEq. (2).

4.3. Welfare

There are two dimensions to consider in the welfare analysis of re-distributive taxation: equity (or related notions of distributive justice)and efficiency. There is a tradeoff between these two dimensions, andboth are jointly determined by the tax rate in the political sector andthe labor supply decisions made in the economic sector. Thus, thewelfare analysis must consider the combined political economyeffects in the two sectors. The tradeoff is explicitly modeled in the the-oretical framework we use: the more pre-tax income is going to beredistributed, the less laborwill be supplied. Assuming that eachworkerchooses his labor supply optimally given the tax rate, we can constructan equity–efficiency frontier, for any particular measure of equity andefficiency. We use one minus the post-tax Gini coefficient as our mea-sure of equity, and total income as the measure of efficiency.17 Usingthese measures, we define the equity–efficiency frontier as the locusof points in this two dimensional space corresponding to after tax

17 There are alternativemeasures aswell, such as the variance of the income distributionto measure inequality or netting out the effort costs of labor in the measure of efficiency.These alternative measures lead to similar conclusions.

equity–efficiency pairs that would arise from optimal labor supplybehavior as we vary tax rates from 0 to 1. We use this as a benchmarkwith which to compare the actual equity–efficiency tradeoff that isobserved in the experiment.

Fig. 2 displays all the equity–efficiency pairs for all group outcomesin the low inequality and high inequality treatments, respectively. Thesolid line in the figures marks the frontier with the upper left of thefrontier corresponding to t=1 and the lower right of the frontier corre-sponding to t=0.18 Table 6 below compares the theoretical equilibriumgini coefficient (equity) and total income (efficiency) with the aver-ages across all the equity–efficiency pairs, separately for the twoinequality treatments, with group-clustered standard errors in pa-rentheses. There is no significant difference between the theoreti-cal equity–efficiency pairs and the observed means except forefficiency in the high inequality treatment (p b 0.05), which is belowthe theoretical level and consistent with the labor supply findingsreported in Table 5.

From a slightly different perspective, Fig. 3 displays total taxrevenues as a function of the tax rate t. The solid line representsthe theoretical Laffer curve, derived under the assumption that allagents supply labor optimally given the tax rate, while the dataobserved in the experiments are marked as the circles. This graphis essentially a different projection of the three-dimensional picturethat summarizes the relation between tax rates, efficiency andequality in the economy.

These findings can be summarized as follows. First, Fig. 2 shows that,consistent with the theoretical equity–efficiency tradeoff, higher taxrates lead to lower aggregate labor supply and lower total income inboth inequality treatments, in exchange for a reduction in after-tax in-come inequality. Second, deviations from the theoretical frontiers inLow inequality treatment are minimal, balanced between pointsabove and below the frontiers and are not correlated with the tax rate(as seen from the right panels of Figs. 2 and 3). Third, in the high in-equality treatment deviations from the theoretical efficiency–equityfrontier are significant. Fourth, as shown in Table 6, these deviationsare entirely along the efficiency dimension, and with no significantdifference in the equity dimension; furthermore, Table 5 showsthat this is driven by the undersupply of labor by the highest produc-tivity agent (that with productivity wi = 35). The figures also showthat this undersupply does not seem to depend on the tax rates: un-dersupply of aggregate labor is observed for both high and low taxrates.

Result 3.

(a) Higher tax rates lead to lower total income irrespectively ofthe inequality level, as predicted by the efficiency–equitytradeoff.

(b) In the low inequality treatment, the data closely track the effi-ciency–equity frontier and theoretical Laffer curve, while inthe high inequality treatment the deviations from these fron-tiers are significantly negative in the efficiency dimension.

5. Extensions: political institutions and social preferences

5.1. Comparison of direct democracy and representative democracy

In this sectionwe compare how two political regimes, direct democ-racy (DD) and representative democracy (RD), affect implemented taxrates, voting behavior, and labor supply.

18 The frontier as we have defined it does not represent the boundary of feasible equity–efficiency pairs. In principle, workers are free to supply 25 units of labor for any tax rate,but doing so is not consistent with equilibrium in our labor market.

Fig. 2. Equity–efficiency frontier.

52 M. Agranov, T.R. Palfrey / Journal of Public Economics 130 (2015) 45–58

5.1.1. Implemented taxesTable 7 summarizes the implemented tax rates in each political

regime in each inequality treatment. The results of a regressionanalysis confirm that in each political regime, higher inequalityleads to significantly higher level of redistribution.19 Moreover,there is no significant difference between the average or medianimplemented taxes in the DD and RD regimes, for each inequalitytreatment.20

5.1.2. Voting behaviorBesides the predictions about equilibrium tax rates as a function of

the distribution of wage rates, the model also makes more specific pre-dictions about voter behavior in the two institutional regimes. Specifi-cally, in DD, all voters, regardless of productivity, have a dominantstrategy to propose their most preferred tax rate, assuming all voterssupply labor optimally conditional on any tax rate. Similarly, in RD,voters have a dominant strategy to vote for the candidatewho proposedthe more preferred of the two candidates' tax rates, once again underthe assumption that all voters supply labor optimally. Moreover, inboth political regimes, theory predicts that ideal tax rates of agents aremonotonic in agents' productivities.

We test these predictions by comparing ideal tax rates of agents pre-dicted by theory with median taxes proposed by experimental subjectsin the DD treatment andwith empirically estimated ideal tax rates fromthe RD treatment. Estimation of empirical ideal taxes in RD regime uti-lizes voting data and backs out a tax rate which minimizes the utilityloss from the voting mistakes for each productivity separately in eachinequality treatment. To define the utility loss from voting mistakes,we first organize voting data in the followingway. For each productivityand each inequality treatment separately, we display simultaneously

19 For each political regime, we regress implemented taxes on a dummy variable for thehigh inequality treatment, clustering observations by group. Estimated coefficients arepositive and significant: β = 0.21 (p b 0.01) for DD regime and β = 0.28 (p b 0.01) forRD regime.20 To reach this conclusion we regress implemented taxes on a dummy variable for theRD regime, for each inequality treatment separately, clustering observations by group.The estimated coefficient is not statistically different from zero for either inequality treat-ment:β=0.06 (p=0.25) in the high inequality treatment and β=0.01 (p=0.95) in thelow inequality treatment. This is further confirmed by aWilcoxon Rank-sum test that failsto reject the null hypothesis that distributions of implemented taxes are the same in theDD and RD regimes, for both the high and low inequality treatments (p=0.06 in high in-equality treatment and p = 0.70 in low inequality treatment).

the two proposals that are offered in each election and the proposalthe voter of that productivity voted for (see Fig. 5 in Appendix B). Thehorizontal axis represents the tax rate proposed by the candidate thevoter voted for, and the vertical axis corresponds to the tax rateproposed by the other candidate. Each panel also has two crossing linesegments. Those line segments represent pairs of tax proposals thatthe voter in theoretically indifferent between. One of the segments,the upward sloping one, obviously is the diagonal. The other, downwardsloping line represents pairs that are equidistant from the voter's idealtax rate. The two lines intersect at the ideal tax rate of the voter. There-fore, correct votes are in north and south quadrants. Incorrect votes arein the east and west quadrants.21 Utility loss is defined as the shortestdistance from a point in the east or west quadrant to the closestindifference line.

Fig. 4 summarizes theoretically predicted and empirically estimatedideal tax rates of agents in both political regimes and both inequalitytreatments. While there are some minor discrepancies betweenempirical and theoretical ideal taxes,22 our data in both political regimesclearly supports the main prediction, the monotonicity result. In bothpolitical regimes, empirical ideal taxes are monotonic in agents'productivities.

A final qualitative theoretical prediction about proposals in DDregime is that the median proposal will be submitted by the medianproductivity type. While this does not always occur in our data, it isthe modal observation: the median type submits the median proposal52% of the time in DD-high treatment 58% of the time in DD-lowtreatment.

5.1.3. Labor supplyThe labor supply behavior in both political regimes is very similar as

can be seen in Table 12 in Online Appendix 3, in which we reports

21 For high productivity voters whose ideal point is zero tax, the west and south quad-rants do not exist, reflecting the fact that it is always optimal for these voters to vote forthe lower tax rate.22 In the low inequality treatment, the ideal tax rates of two lowest productivity agentsare not perfectly ordered by productivity in DD regime, but neither one is significantly dif-ferent from the theoretical one, nor are they significantly different from each other. Sec-ond, average proposed taxes by the highest productivity workers are above 0.00. This isexpected, since any variation in behavior of these types produces this result, as negativetaxes are not allowed. The positive tax rate proposals by the high types is also consistentwith finding by Feddersen et al. (2009) who report that voting behavior ismore generousto others, the less likely a voter is pivotal. In DD-high, proposals by the high productivitytypes are pivotal only 7% of the time.

Table 6Equity–efficiency tradeoff.

Equilibrium Mean observed (std err)

High inequality treatmentTax rate 0.53 0.50 (0.03)Gini coefficient 0.31 0.32 (0.019)Total group income 899.14 818.32 (40.99)

Low inequality treatmentTax rate 0.28 0.26 (0.03)Gini coefficient 0.35 0.35 (0.015)Total group income 512.16 518.98 (16.40)

Note. Robust standard errors in parentheses, clustered by group.

53M. Agranov, T.R. Palfrey / Journal of Public Economics 130 (2015) 45–58

analysis parallel to Table 4 conducted separately for each politicalregime.23

5.1.4. WelfareFinally, we conduct welfare analysis similar to the one reported in

Section 4.3 separately for each political regime. Tables 13 and 14 (OnlineAppendix 3) replicate Table 6,which reported average efficiency–equitymeasurements for each political regime. Figs. 6 and 7 (Online Appen-dix 3) replicate Fig. 2 and plot all the equity–efficiency pairs togetherwith the theoretical benchmark (equity–efficiency frontier). We donot observe any significant aggregate differences between politicalregimes in terms of observed efficiency or equity in both inequalitytreatments. Moreover, both political regimes closely track theoreti-cal predictions.

5.2. Social preferences

5.2.1. Theoretical predictions of model with social preferencesIn this section we extend our theoretical analysis to characterize

how labor supply and equilibrium taxes are affected by socialpreferences.24 This model is a natural one for considering the effectsof other-regarding preferences. Indeed, in this framework socialpreferences of almost any kindwill affect both labor supply decisionsof agents as well as their indirect preferences over tax rates. Weexplore in depth two commonly used social preferences models:altruism and inequity aversion.

5.2.1.1. Preferences for altruism. UiA(w, x, t) denotes the altruistic utility

function of agent i:

UAi w; x; tð Þ ¼ Ui wi; xi; tð Þ þ A

1n−1

Xj≠i

U j wj; xj; t� �

where parameter A ≥ 0 measures i's altruism, the weight i puts on theaverage payoff of others in the society, and Ui(wi, xi, t) is defined asbefore by Eq. (1). The standard model without social preferences isnested in altruism model and corresponds to A = 0.

23 Regression analysis similar to the ones reported in Table 5 performed separately foreach political regime suggests that the undersupply of labor by the w = 35 types in theHigh Inequality treatmentwhen the investment constraint was binding (t b 0.36)was sig-nificant in the DD treatment, but not significant in the RD treatment. Further, in the lowinequality treatment, the slope of the lowest productivity type's normalized labor supplyis significantly less than the predicted value of 0.8. However, this has very little economicsignificance, as this agent type is predicted to supply very little labor for any tax rate com-pared to other agent types.24 All proofs of results in this section are available in Online Appendix 1.

5.2.1.2. Fehr–Schmidt preferences. Order agents according to theirproductivity from the lowest i = 1 to the highest i = n. Ui

FS(w, x, t)denotes the Fehr–Schmidt utility function of agent i:

UFSi w; x; tð Þ ¼ Ui wi; xi; tð Þ −

αn−1

�Xnj¼1

max U j wj; xj; t� �

−Ui wi; xi; tð Þ;0� �−

−β

n−1�Xnj¼1

max Ui wi; xi; tð Þ−U j wj; xj; t� �

;0� �

where the second term measures utility loss from disadvantageous in-equality in payoffs and the third termmeasures utility loss from advan-tageous inequality in payoffs. The standard assumption in the literatureis that 0 ≤ β ≤ α ≤ 1, i.e. individuals experience greater utility loss frominequality when their payoff is below average than when their payoffis above average. The standard model without social preferences isnested in the inequity aversion model and corresponds to α = β = 0.

5.2.1.3. Labor supply with social preferences. Proposition 2 below estab-lishes two results. First, altruism leads to higher labor supply comparedto the selfish model, for all values of A and for all productivity levels.Second, inequity aversion leads to higher individual labor supply ifand only if an individual's productivity rank is sufficiently high. Thus,for any tax rate, relatively high productivity inequality averse individ-uals will supply more labor than a selfish individual, while relativelylow productivity workers will supply less labor.

Proposition 2. Labor supply.

1. The optimal labor supply of an agent with productivity wi andaltruism parameter A is xAi ðwi; tÞ ¼ ½1−t þ ð1þAÞt

n �wi . The moreagent i cares about the average payoff of other agents the morelabor he will supply for a given tax rate t. That is, xiA(wi, t) N xi⁎(wi, t)and dxAi ðwi ;tÞ

dA N 0 f or all A≥0.2. The optimal labor supply of an individual with productivity wi and

Fehr–Schmidt parameters α and β is: xFSi ðwi; tÞ ¼ ð1−t þ 1

μ i� tnÞwi

where μ i ¼ 1þ αðn−iÞ−βði−1Þn−1 N0 for all 1 ≤ i ≤ n. Inequality averse

agents with high productivity, those with iN αnþβαþβ , supply more

labor than their selfish counterparts, while inequality averse agentswith low productivity, those with i≤ αnþβ

αþβ supply less labor thantheir selfish counterparts.

5.2.1.4. Equilibrium tax effects of social preferences. Altruism and inequityaversion affect equilibrium taxes in opposite ways. A society of altruistsprefers lower taxes, as everyone is concerned, at least to some degreewith efficiency, which declines when taxes increase. Thus, a small in-crease in altruismwill result in a lower ideal tax rate for each individual(unless the individual's ideal tax ratewas already equal to 0).With ineq-uity aversion, because μm N 1, at the margin the median voter is moreconcerned about reducing the payoff of higher productivity workersthan increasing the payoff of lower productivity workers, even if thismeans lowering her own payoff. There are some minor second andthird effects that can go the other way, under fairly weak conditionsthose other effects are small enough for the main intuition to hold.

Proposition 3. Tax rates. Assume n N 3 and w2mb

1n Z.

1. The ideal tax rate of the altruistic median productivity agent with0 b A ≤ 1 is lower than that of the selfish agent with A = 0, i.e.tmA b tm⁎.

2. If individuals have inequality averse preferences such that 0 b

β ≤ α ≤ �αðβ;nÞ, then the ideal tax rate of themedianproductivity in-dividual is strictly higher than that individual's ideal tax rate wouldbe if α = β = 0, i.e. tmFS ≥ tm⁎ .25

25 The sufficient condition α≤ �αðβ;nÞ is stronger than needed. For n sufficiently large, itcan be dispensedwith entirely because the second order terms vanish. For the experimen-tal parameters, the condition reduces to α ≤ 3β. See the online appendix for details.

Fig. 3. Laffer curves.

Table 7Implemented taxes in each regime.

High inequality, t⁎ = 0.53 Low inequality, t⁎ = 0.28

Mean (st err) Median Mean (st err) Median

DD 0.47 (0.04) 0.50 0.26 (0.03) 0.25RD 0.54 (0.03) 0.57 0.27 (0.07) 0.23

Note. Robust standard errors are in parentheses, clustered by group.

54 M. Agranov, T.R. Palfrey / Journal of Public Economics 130 (2015) 45–58

In principle, one could extend the analysis above to allow for hetero-geneous social preferences. For the altruism model all the results ofPropositions 2 and 3 continue to hold with heterogeneity. Altruisticagents will work harder and prefer a higher tax rate than selfish agents,regardless of the distribution of altruistic preferences in the population.The implications of heterogeneity for inequity averse agents is muchmore complicated because the impact of social preferences on labor

Fig. 4. Predicted and empirical ide

supply and ideal tax rates would now be two dimensional problem,depending on both worker productivity and individual inequity aver-sion. One possible way to build in heterogeneity is to allow for twotypes of agents, selfish and inequity averse with parameters (α, β).We conjecture that in such a mixture-of-types model the main re-sults would go through: in the labor market, lower income inequityaverse workers will supply less labor and higher income inequityaverse workers more labor; and the median ideal tax rate will behigher compared with the case of selfish preferences. However, afull analysis of this and other theoretical extensions to explicitlyallow for heterogeneity of inequity aversion are beyond the scopeof the present paper.

5.2.2. Estimating social preference modelsIn this section we address the question of whether our data provide

evidence of the presence of other-regarding preferences in the labormarket and/or in the political domain. As observed in Section 4, we

al tax rates, by productivity.

Table 8Estimation results for altruism and inequity aversion effects on labor supply.

Treatment Altruism (A) Inequity aversion (α, β)

High inequality 0 (0,0)Low inequality 0.05 (0,0)

Notes. Data pooled across DD and RD regimes.

Table 9Estimation results for altruism and inequity aversion effects on DD voting.

Treatment Altruism (A) inequity aversion (α, β)

High inequality 0 (0.01,0)Low inequality 0 (0,0)

Table 10Regressions of predicted against observed tax rates.

Constant Slope R2

Theory −0.00 (0.07) 0.94 (0.17) 0.28EE1 0.23⁎ (0.06) 0.12⁎ (0.13) 0.31EE2 0.23⁎ (0.07) 0.11⁎⁎ (0.13) 0.31EE3 0.12⁎ (0.06) 0.32⁎⁎ (0.14) 0.35

Notes. Robust standard errors are in the parentheses, clustered by group.⁎ Is significantly different from 0.⁎⁎ Is significantly different from 1.

55M. Agranov, T.R. Palfrey / Journal of Public Economics 130 (2015) 45–58

cannot reject the hypothesis that labor supply choices and implementedtaxes are the same as the one predicted by the theory with selfishagents. However, the question remains whether our data might alsobe compatible with other-regarding preferences, based on the resultsdeveloped above.

5.2.2.1. Labor supply. For each inequality treatment, we estimate thealtruism parameter A ≥ 0 by finding the value of A that minimizesthe sum of squared deviations of the observed labor decisionsfrom the theoretically predicted ones for that value of A ≥ 0. Weuse a similar method to estimate the two inequity aversion pa-rameters, (α ≥ 0, β ≥ 0). Table 8 presents the results of thisestimation.

With respect to altruism, we measure a positive but statisticallyinsignificant (p = 0.44) altruism parameter (A = 0.05) only in thelow inequality treatment. In all other cases, we also cannot reject thenull that A = 0 or α = β= 0.26

5.2.2.2. Proposal behavior. Social preferences would also affect the pref-erences of agents over the tax schedules. The data from the DD regimeis perfectly suited for testing it because themedianproposalmechanismis specifically designed to directly elicit each voter's ideal tax rate. Sim-ilar to what we did for the labor supply data, we estimate the altruismparameter, A ≥ 0, and the inequity aversion parameters, (α ≥ 0, β ≥ 0),by minimizing the sum of square distances between the observed taxproposals and the optimal tax proposals for each productivity type,conditional on the parameters. The results are reported in Table 9. Weestimate a very small and insignificant (p = 0.48) envy parameter(α = 0.01) for the high inequality treatment. In all other cases thebest fitting parameters equal 0.

5.3. Empirical equilibrium

The results so far paint a picture of the aggregate data as beingclose to the theory based on selfish preferences with respect to(1) the qualitative comparative statics; (2) average and median im-plemented tax rates; (3) the individual labor supply responses totax rates (except for the wi = 35 workers); (4) the aggregate labor

26 The p-value is based on aWald test for whether the sum of squared differences at theestimated parameters are significantly less than the sum of squared differences for theselfish model, with all parameters set equal to 0. The estimates are constrained to benon-negative, in accordance with the theoretical model, so p-values are not reported incases where the sum of squared differences is minimized at the null values of the param-eters. In these cases the unconstrained estimates would be negative.

supply effect of taxes; and (5) voting behavior. However there issome heterogeneity across groups in the data. In this section wetake a closer look at this variation, and in particular explore the pos-sibility that deviations from the equilibrium tax rates may be drivenby variation across groups with respect to expectations about laborsupply responses to taxes.

Theoretically, deviations from equilibrium labor supply responses totax rates, if correctly anticipated by voters, will lead to distortions in thepolitical equilibrium tax rates. That is, the equilibrium tax rates in highand low inequality treatments derived in Section 2 were based on theassumption that all agents make optimal labor decisions at all taxrates, and all voters correctly anticipate this. However, to the extentthat we find actual aggregate labor supply functions to be differentfrom the theoretical ones, if these deviations vary systematically acrossgroups, then one might expect rational candidates to propose differenttax rates in the RD regime and agents to offer different tax proposalsin the DD regime. Therefore, in this sectionwe will connect the analysisof the labor and political markets and ask whether the variation in thelabor supply across different groups is linked in this way to the variationin the implemented tax rates. We refer to tax rates that constitute anequilibrium relative to the empirical labor supply functions as anempirical equilibrium.

To do this, we construct three alternative models of “empiricalequilibrium” (EE) tax rates that differ according to the methodused to estimate the labor supply functions in a group. That is, we es-timate empirical labor supply functions of each agent in each group,and then compute the empirical equilibrium tax rate for that groupbased on the estimated labor supply functions. The challenge is toobtain good estimates of the labor supply functions. To deal withthis issue, rather than choosing one particular method to estimatelabor supply, we apply three different alternative models to do thisestimation. The first, EE1, uses only the data from the first 10 periodsto estimate the labor supply functions of each group member, anduses this estimate to compute an adjusted median voter's ideal taxrate as the basis for the empirical equilibrium tax rate. The second,EE2, is similar, but uses the labor supply data from all 20 periods.The third model, EE3, takes a different approach. For each groupEE3 is based only on the earnings of the median productivity workeracross the ten trial tax rates in the first 10 periods; the EE3 tax rate isthe one of these for which that agent experienced the highestearnings.

Table 10 shows the results of regressing predicted against observedtax rates, using all 400 observations from the 40 groups in the experi-ment. The observed tax rate in each group equals the median of thatgroup's ten implemented taxes in periods 11–20. The predicted taxrate is calculated for each of the models EE1, EE2 and EE3 describedabove, aswell as for the theoreticalmodel based on individually optimallabor supply, derived in Section 2.

The first model, based on the theoretical labor supply functions,is the only model that produces estimates close to the theoreticalpredictions. The first model has an estimated intercept equal to0.00 and slope equal to 0.94, and we cannot reject the hypothesis

56 M. Agranov, T.R. Palfrey / Journal of Public Economics 130 (2015) 45–58

that they equal 0 and 1, respectively. Based on the coefficient esti-mates, all three EE models reject that hypothesis. In fact, for EE1and EE2, one cannot even reject the hypothesis that the slope equals0. In terms of model fit, the R2 is slightly higher for the three EEmodels than the theoretical equilibrium model, but this does nottake into account that we are implicitly burning some degrees offreedom by estimating the labor supply curves for each group andthen feeding those estimates into each of the EE models. We inter-pret this finding as supportive of the basic theory. That basic theoryobviously cannot explain variation in the data across groups, but theresults in table show that on average its overall predictive accuracyis very good. In a sense, this just mirrors the results reported inTables 3 and 7 and Fig. 1, showing that average tax rates in thetwo inequality treatments are very close to their predicted values.The additional estimation to fit each group separately based ontheir own empirical labor supply functions, while explaining someof the variance across groups, does so at a cost in predictiveaccuracy.

6. Conclusion

This article presents the results from an experiment to explorethe median voter theory of equilibrium income tax rates that pro-duce distortions in labor supply. The experiment is novel in a num-ber of ways, including combining a labor market with a politicalmarket, where preferences in the political market are endogenousand are determined by expectations about labor supply responsesto taxes. The central focus was on four main questions. Does greaterinequality ex ante lead to higher tax rates and more income redistri-bution? Are the implemented tax rates driven by the induced prefer-ences of the median income voter? Do the implemented tax ratesdepend on the institutional rules governing the collective choiceprocedure? Do social preferences have a significant impact on laborsupply responses to tax rates or to indirect voter preferences overtax rates?

The answer to the first question is unambiguously yes. Higherex ante inequality in terms of worker wage rates leads to highertax rates. The effect is significant and large in magnitude. The an-swer to the second question is related to the first: the implement-ed tax rates in both inequality treatments are almost exactly equalto the theoretical ideal tax rate of the median wage worker. Theanswer to the third question is negative. We do not observe anysignificant differences in labor supply or average implementedtax rates between the direct democracy institution and a represen-tative democracy where tax rates are determined by candidatecompetition. While there are many other possible democratic col-lective choice procedures that one could examine, this third find-ing is at least suggestive of a robustness with respect to the finerdetails of majoritarian democratic choice procedures. The answerto the fourth question is also negative. We do not observe signifi-cant deviations from labor supply behavior or voting behavior ofthe sort that are implied by models of altruism or inequity aver-sion. The one exception is the labor market behavior under theDD-low treatment, where we estimate a small but significant al-truism effect.

The findings from the experiment lead to some strong conclu-sions, but leave open a number of more difficult questions that arebeyond the scope of the analysis presented here. There are at leasttwo intriguing unanswered questions about behavior in these ex-periments. First there is the surprising result about undersupply oflabor by the highest productivity workers in the high inequalitytreatment. Such behavior is inconsistent with selfish behavior aswell as altruistic or inequality averse behavior. Second, we observesome variation in tax rates across different groups and acrossperiods.

More general questions concern the robustness of our find-ings to richer environments. The findings are suggestive of rathergeneral phenomena, but as a first exploration of these phenome-na in the laboratory, our experimental environment was neces-sarily very stark. What happens if there are more agents, morecomplicated political institutions involving multiple layers andbranches of government, progressive tax structures, or dynamicconsiderations such as income mobility or investment inhuman capital? The taxes we consider are purely redistribution-al, but many government expenditures are not purely redistribu-tional and involve investments in public infrastructure, socialinsurance, and other categories that have a significant publicgood component. There are also interesting questions about theeffect of tax rates on tax compliance, an issue that is beyond thescope of the present study. All of these issues are important tounderstand the relationships between public finance and politi-cal economy more deeply, and some of them are already beingexplored theoretically and empirically. We are hopeful that thispaper opens the door for further investigation of these issuesusing laboratory experiments as a complement to theoreticaland empirical studies.

Appendix A. Proof of Proposition 1

Proof of Proposition 1. Single-peakedness is established in two steps.Clearly, if d2U�

i ðwi ;tÞdt2

b 0 in the region t ∈ 0, 1], then single peakedness inthe policy space follows immediately. From Eq. 3, we get:

dU�i wi; tð Þdt

¼ −w2i 1−t þ t

n2

� �þ 1nZ 1−2

n−1n

t� �

d2U�i wi; tð Þdt2

¼ n−1n2 nþ 1ð Þw2

i −2Z� �

:

Thus, single-peakedness is guaranteed by concavity ofUi⁎ for all indi-

vidualswhose productivity is sufficiently low (w2i b 2

nþ1 Z). For relativelyhigh productivity workers (w2

i N2

nþ1 Z), it is easy to show that dU�i ðwi ;tÞdt b 0

for all values of t ∈ 0, 1]:

−w2i 1−t þ t

n2

� �þ 1nZ 1−2

n−1n

t� �

b0⇔w2i NZ � n 1−2tð Þ þ 2t

n2 1−tð Þ þ t:

The last inequality is satisfied for all w2i N 2

nþ1 Z since 2nþ1 N

nð1−2tÞþ2tn2ð1−tÞþt .

From the above argument, if w2i N

1n Z then the ideal tax rate of the

worker with productivity wi is zero. However, if w2i ≤

1n Z , then the

ideal tax rate of worker with productivity wi, denoted by ti⁎, can bewritten as

t�i ¼n2

n2−1�

1nZ−w2

i

2nþ 1

Z−w2i

:

It is easy to check that these ti⁎ are weakly monotone in the produc-tivities. The third property follows immediately. QED

57M. Agranov, T.R. Palfrey / Journal of Public Economics 130 (2015) 45–58

Appendix B. Voting in RD

Fig. 5. Voting behavior in RD treatments.

Appendix C. Supplementary data

Supplementary data to this article can be found online at http://dx.doi.org/10.1016/j.jpubeco.2015.08.008.

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